All we know is that will always be 1, it requires a little bit more to assert that or .

To show why this is the case, consider a parametrization for , with normal . For a second parametrization with normal , we will have , where is the Jacobian. Therefore the normal vectors to at , will either be equal or opposite, depending on the sign of the Jacobian. This means
the Gauss map will not alter if J>0, and will only change sign if J<0.